Kinetic limit of a conservative lattice gas dynamics showing long-range correlations

An anisotropic lattice gas dynamics is investigated for which particles on ^d jump to empty nearest neighbor sites with (fast) rate ^–2 in a specified direction and some particular configuration-dependent rates in the other directions. The model is translation and reflection invariant and is particle conserving. The space coordinate in the fast-rate direction is rescaled by ^–1. It follows that the density field converges in probability, as 0, to the corresponding solution of a nonlinear diffusion-type equation. The microscopic fluctuations about the deterministic macroscopic evolution are determined explicitly and it is found that the stationary fluctuations decay via a power law (1/r ^d ) with the direction dependence of a quadrupole field.     

Superdegenerate point in FCC phase diagram: CVM and Monte Carlo investigations

We investigate the topology of the phase diagram of binary alloys on the fee lattice with first-neighbor antiferromagnetic interactions around the superdegenerate point, where the L1₀ and L1₂ phases meet. We treat the system as a hard-constraint lattice gas, following a procedure previously described by Lebowitzet al. We perform cluster variation method calculations in theT0 limit and Monte Carlo simulations directly atT=0 K on the ground states of the superdegenerate point. We find that: (i) there is no disordered phase in the neighborhood of this point; (ii) a phase L for which two of the four cubic sublattices have the same average occupancy and each of the two others are different appears between L1₀ and L1₂; (iii) the transition L/L1₂ is of first order.     

A fixed-point equation for the high-temperature phase of discrete lattice spin systems

A fixed-point equation on an infinite-dimensional space is proposed as an alternative to the usual definition of the infinite-volume limit in discrete lattice spin systems in the high-temperature phase. It is argued heuristically that the free energy and correlation functions one obtains by solving this equation agree with the usual definitions of these quantities. A theorem is then proved that says that if a certain finite-volume condition is satisfied, then this fixed-point equation has a solution and the resulting free energy is analytic in the parameters in the Hamiltonian. For particular values of the temperature this finite-volume condition may be checked with the help of a computer. The two-dimensional Ising model is considered as a test case, and it is shown that the finite-volume condition is satisfied for0.77 _critical.     

Hamiltonian limit of the 3D Zamolodchikov model

A two-dimensional quantum Hamiltonian _N,M commuting with the layer-to-layer transfer matrix of the three-dimensional Zamolodchikov model is derived. This Hamiltonian is defined on a lattice ofN×M sites. The special casesN×2, 2×M, and 3×M are studied.This paper is dedicated to Cyril Domb.     

On the finite-size scalling equation for the spherical model

The mean spherical model with an arbitrary interaction potential, the Fourier transform of which has a long-wavelength exponent , 0<2, is considered under periodic boundary conditions and fully finite geometry ind dimensions, when <d<2. A new form of the finite-size scaling equation for the spherical field in the critical region is derived, which relates the temperature shift to Madelung-type lattice constants. The method of derivation makes use of the Poisson summation formula and a Laplace transformation of the momentumspace correlation function.On leave of absence from Institute of Mechanics and Biomechanics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria.     

Free energy of the solvable chiral Potts model

Very recently, it has been shown that there are chiralN-state Potts models in statistical mechanics that satisfy the star-triangle relation. Here it is shown that the relation implies that the free energy (and its derivatives) satisfies certain functional relations. These can be used to obtain the free energy: in particular, we expand about the critical case and find that the exponent is 1–2/N.     

Defect concentration in Ti-substituted YIG from TG curves

The article discusses theory and experiment about the measurement of defect concentration in Ti-substituted yttrium iron garnet by means of thermogravimetry techniques. The two possible cases arising from oxygen interchange with atmosphere, oxygen vacancies and interstitial cations, may be analyzed quantitatively from the derived expressions. The possibility of another type of defects being present in the samples, not associated to oxygen evolution, is not excluded. Measurements were carried out in air and CO₂ for different Ti contents. The substitution tends to increase slightly the defect parameter θ in YIG, while a CO₂ reducing atmosphere is much more effective than the Ti substitution for increasing θ. Calculations for a given single-phased sample sintered in air show two possibilities: a deficit of one oxygen atom for every 2.5 unit cells of 96 atoms, or one excess cation for every 3.5 cells. Other samples show very similar results. The accuracy involved in the measurements is about 2–3%.     

Dynamic percolation transition induced by phase separation: A Monte Carlo analysis

The percolation transition of geometric clusters in the three-dimensional, simple cubic, nearest neighbor Ising lattice gas model is investigated in the temperature and concentration region inside the coexistence curve. We consider quenching experiments, where the system starts from an initially completely random configuration (corresponding to equilibrium at infinite temperature), letting the system evolve at the considered temperature according to the Kawasaki spinexchange dynamics. Analyzing the distributionn _l(t) of clusters of sizel at timet, we find that after a time of the order of about 100 Monte Carlo steps per site a percolation transition occurs at a concentration distinctly lower than the percolation concentration of the initial random state. This dynamic percolation transition is analyzed with finite-size scaling methods. While at zero temperature, where the system settles down at a frozen-in cluster distribution and further phase separation stops, the critical exponents associated with this percolation transition are consistent with the universality class of random percolation, the critical behavior of the transient time-dependent percolation occurring at nonzero temperature possibly belongs to a different, new universality class.     

Equilibrium polymerization in a solvent: Solution on the Bethe lattice

The lattice model for equilibrium polymerization in a solvent proposed by Wheeler and Pfeuty is solved exactly on a Bethe lattice (core of a Caylay tree) with general coordination numberq. Earlier mean-field results are reobtained in the limitq, but the phase diagrams show deviations from them for finiteq. Whenq=2, our results turn into the solution of the one-dimensional problem. Although the model is solved directly, without the use of the correspondence between the equilibrium polymerization model and the diluten0 model, we verified that the latter model may also be solved on the Bethe lattice, its solution being identical to the direct solution in all parameter space. As observed in earlier studies of the puren0 vector model, the free energy is not always convex. We obtain the region of negative susceptibility for our solution and compare this result with mean field and renormalization group (-expansion) calculations.     

Infrared asymptotic freedom of a hierarchical φ 3 6 lattice theory

For the weakly coupled lattice ₃ ⁶ theory in a hierarchical model approximation a nonperturbative renormalization group analysis in the spirit of Gawedzki and Kupiainen is performed to study the flow of the effective actions. We deduce a domain of attraction to the tricritical (Gaussian) fixed point. The two relevant coupling constants of the problem are controlled by analytic continuation to complex domains, tracing their images under the renormalization group iterations.